Classical Optimal Replacement Strategies Revisited - IEEE Xplore

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IEEE TRANSACTIONS ON RELIABILITY, VOL. 65, NO. 2, JUNE 2016

Classical Optimal Replacement Strategies Revisited Maxim Finkelstein, Mahmood Shafiee, and Anselme N. Kotchap

Abstract—This paper considers the generalization of the classical optimal age replacement strategy to the case when the system's output is modelled by the decreasing deterministic function. This function describes an additional source of system's deterioration with time. We derive and analyze the long-run expected cost per unit of time and consider the corresponding optimal replacement problem. Our results show that additional source of degradation decreases the optimal replacement time as compared with the case without output or with a constant in time output. Furthermore, the optimal replacement time can now exist and be finite when the failure time of our system is described by the exponential, non-aging distribution. The cases of periodic replacement and of stochastic output are also considered and analyzed. Some simple examples illustrating our results are given.

Cost of replacement.

Index Terms—Degradation, output function, renewal reward process, replacement.

Navigation error.

Maximal penalty. Cost of minimal repair. . Stochastic output. Univariate pdf of stochastic output. CDF that corresponds to

.

Univariate conditional pdf of stochastic output. CDF that corresponds to

.

Function of a random parameter PDF of

NOTATION

.

.

First passage time for the gamma process.

Output function. . Initial level of output.

I. INTRODUCTION

Stochastic output

R

Time to failure of a system. CDF of

. .

PDF of

.

Failure rate that corresponds to

.

Renewal function. Expected level of the output. Long-run expected cost per unit of time. Expected duration of the renewal cycle for age replacement. Cost of repair on failure.

Manuscript received September 25, 2014; revised March 03, 2015; accepted December 23, 2015. Date of publication January 27, 2016; date of current version May 30, 2016. Associate Editor: D. Theilliol. M. Finkelstein is with the Department of Mathematical Statistics, University of the Free State, 339 Bloemfontein 9300, South Africa, and also with ITMO University, St. Petersburg 197101, Russia (e-mail: [email protected]). M. Shafiee is with the Institute for Energy and Resource Technology, School of Applied Sciences, Cranfield University, Bedfordshire, MK43 0AL, U.K. (e-mail: m.shafiee@cranfield.ac.uk). A. N. Kotchap is with the Department of Mathematical Statistics, University of the Free State, 339 Bloemfontein 9300, South Africa. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TR.2016.2515591

ELIABILITY literature mostly focuses on modelling the operation of binary or multistate systems with discrete levels of output. Less attention has been devoted to reliability analysis of systems with continuous output levels [1]. However, output can be considered as an important overall characteristic of a quality and should be included in stochastic description of reliability indices [2]–[4]. By the output function, we understand in this paper, in fact, the corresponding rate, e.g., the production in a unit interval of time. For instance, consider an electrical power generating system (e.g., a steam power turbine), which is a rather complex technical system. Its performance at each instant of time is defined by the output power. It is impossible (and there is no theory for that) to define the output power as a function of states of the components. However, it is a crucial overall characteristic of performance that exists and well measured and can be the basis for different reliability-wise decisions including optimal maintenance strategy. Due to deterioration, the output of technical systems (or its expectation) is usually a decreasing function of time. For instance, as follows from [5], the power of a steam power turbine generating unit is decreasing in expectation. This is obviously the case for other types of turbines and for a variety of other complex technical systems as well. Obviously, the cumulated (integrated) output is increasing in time. Degradation is a complex multidimensional process. Reliability-wise, it is often manifested by the increasing failure rate of a system, whereas the real mechanism that results in this phenomenon is hidden. However, the decreasing output of systems

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FINKELSTEIN et al.: CLASSICAL OPTIMAL REPLACEMENT STRATEGIES REVISITED

with non-binary outputs is also a characterization of degradation. At many practical instances, this degradation can be considered as independent from stochastic degradation that results in the increasing failure rate or in the other aging pattern. Thus these two processes can describe degradation of different nature. For instance, the process of gradual ‘misalignment’ of different parts of a system results in the decrease in the output, whereas a system failure occurs when one of the parts of the system experiences a hard failure. In accordance with the above reasoning, in this paper, we assume, that the deteriorating output of a system during operation is modelled by a positive, decreasing deterministic or random function, whereas the hard failure mechanism is ‘independent’ of this process. This is a reasonable assumption that captures the main features of the model for further optimization of the maintenance activities. We point out that, under the assumption of additivity, the integral of the output function can be viewed as some reward in a given interval of time and, therefore, our optimal replacement problems can be also considered in the light of the renewal-reward processes. The main goal of this paper is to generalize the classical PM strategies in [6], which are traditionally justified by the increasing failure rate of a system to the case of additional source of deterioration defined by the decreasing output. Therefore, the finite optimal time of age replacement can exist, e.g., for systems with constant and even decreasing failure rate (see Remark 1). The derivations of the main results in Section III are rather straightforward; however, we believe that the main contribution of this note is in the meaningful practical setting and its stochastic interpretation. Mathematical reasoning, as in the original paper [6], is fairly simple, however, as far as we know, this setting was not addressed in the literature so far. A possible way of generalization is to consider bivariate optimization with parameters of age and the level of output as the corresponding variables. However, this would be more in the frame of the condition-based maintenance and can constitute a topic for further studies. This paper is organized as follows. In Section II, some supplementary results [1], [7] are obtained that help to formulate our setting and approach for the corresponding optimal replacement problems in Sections III and IV. We briefly discuss the stochastic output case in Section V and end with concluding remarks in Section VI.

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, the survival function by , the corresponding probability density function (pdf) by , and the failure rate by . In reality, the relation between the aging properties of (e.g., the increasing failure rate) and the decreasing property of can be rather complicated, and we can say that both of them (if present) are different manifestations of the manifold degradation processes that evolve in a system. Thus, the failure times of our system constitute a renewal process. Assume that the cumulated output in is , then the corresponding process can be considered as the renewal reward process. Therefore, as mentioned, has the meaning of some rate (e.g., output power for the power generating unit). The non-additive case can be also considered (e.g., the decreasing with time accuracy of some measurement systems) but in this paper, we focus on the additive setting. We are interested first in the expected level of the output of the repairable system at time to be denoted by . Applying the standard reasoning of renewal theory, it can be easily shown [1] (1) is the corresponding renewal density function, the where first term represents the output level at time if no system failures had occurred before time , and the integrand represents the output level at time if the last failure had occurred in the interval with no further failures in the interval . Assume that . For decreasing , the sufficient condition for this inequality is just the existence of the expectation of . Applying the key renewal theorem, the stationary value can be obtained as (2) is the expected length of a renewal where cycle is assumed to be finite. Equation (2) [but not (1)] can be also derived via the renewal reward processes concept [1], [8]

(3)

II. RENEWAL PROCESS WITH CONTINUOUS OUTPUT In this introductory section, we need some supplementary material to be presented mostly following [1]. Denote by the output (rate) of our system in a unit interval of time and assume that it is a continuous, decreasing (non-increasing), deterministic function describing in some ‘aggregated form’ deterioration in performance of the system with time. Let the system be instantaneously and completely (perfectly) repaired on failure, which also means that its output is brought back to the initial level . As discussed in the Introduction, we assume that the time to failure of the system and the described degradation processes are “independent.” Denote its cumulative distribution function (cdf) by

is fixed, and we cannot control it and, therefore, The value consider an optimization problem without introducing preventive maintenance and the corresponding costs for the renewal cycle. This will be done in the next section. Note that (2) can be generalized to the case when the output level is defined by a continuous, decreasing (e.g., in sample paths and, therefore, in the mean value) stochastic process [1] as (4) where

is the expectation (mean) of stochastic process and we assume that in .

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III. OPTIMAL AGE REPLACEMENT POLICY DETERMINISTIC OUTPUT

additional function pital’s rule that

FOR

Consider the problem of minimizing the expected cost per unit of time for an infinite time span. A repairable system is replaced at time from the last renewal point or at failure, whichever comes first. The time of repair is assumed to be instantaneous; however, the setting can be generalized to the non-instantaneous case as well. Let be the cost of repair on failure, which includes the replacement and other costs. Therefore, , where is just the cost of replacement. If a system operates for time without failure or replacement the negative accumulated cost (reward), similar to the previous section, is assigned as , where is a decreasing (nonincreasing) function. Thus, our output function has a sense of reward in a small unit interval of time. For simplicity, we will assume that the reward is expressed in monitory units (the same as cost). We must find that minimizes —the expected cost per unit of time (the cost rate) for an infinite time span. In fact, this is a generalization of the classical optimal age replacement problem of Barlow and Hunter [6] to the case of the continuous output (reward). Hundreds of papers and numerous books have been written on different aspects of optimal maintenance (see e.g., monographs [9], [10] and references therein), however, as far as we know, the described in our paper important generalization was not considered so far. Thus, in accordance with the renewal reward concept, the long run expected cost per unit of time for the described setting is

. It can be seen using the L'Hos-

On the other hand

as the numerator is the limit of the expected profit on the renewal cycle, is a decreasing function, and is just the limit of the duration of this cycle. Here and in what follows, we also assume that

which is, in fact, an obvious condition that is usually met for decreasing . Note that the sign of is defined by the following function:

(6) which is negative as

is the mean reward during the renewal cycle, whereas can be interpreted as the mean reward during the cycle when is constant and is equal to . Thus the cost function is negative and is increasing from to . Therefore, from (5), we obtain (5)

where is the expected duration of the renewal cycle. We see that when , we arrive at (3). Thus, we were able to divide the costs with respect to replacement (failure) costs and the corresponding gain, otherwise analysis would be much more complex if possible at all. Denote the first term in the right-hand side of (5) by and the second term by , i.e., . When , we have: and the initial that corresponds to is just shifted down on units. Obviously, there is no new optimization problem in this trivial marginal case. Assume that the failure rate that corresponds to is increasing. It is well known that , and, therefore, when the function is either decreasing in (no age replacement) or having a single minimum at the finite , such that [7]. Now we have an

Thus, there exists at least one finite or non finite optimal that minimizes in (5). Recall that minimizes the first term and assume that it is finite, i.e., . Then, obviously,

which means that, if there exists the finite To be more precise, we must look at the simple algebra can be transformed to

, then . , which after

(7)


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It is easy to verify via considering the corresponding derivative that the left-hand side of (7) is increasing when, for all , we have (8) which is the case for our assumptions. In order to cross the line and to ensure a single finite , this function (that is equal to 0 at ) should obey the following condition: (9) When , (9) reduces to the well-known condition of Barlow and Hunter [6]. The opposite sign of inequality in (8) means that . Remark 1: Inequalities (8) and (9) prompt us that distinct from the classical case we can have now the optimal value not necessarily under the “deterioration condition” , as we consider an additional source of deterioration manifested by the decreasing . For instance, when is a constant, there is obviously no need for age replacement when considering the classical setting without the output function. However, as (9) holds in the considered case, there exists a finite optimal if

Thus, everything depends on the parameters involved. It can be easily derived using (5) and (7) that when finite, the corresponding cost rate is

Fig. 1. Cost functions for the constant (lower), linear (middle), and zero (upper) output.

VALUES

FOR

TABLE I OPTIMAL TIME AND COST FOR DECREASING OUTPUT

THE

LINEARLY

is VALUES

(10) which is a remarkably simple relationship which reduces to the , i.e., . classical case when Remark 2: In principle, one can describe situations when the output is nonadditive and, therefore, cannot be expressed as some integral of the corresponding decreasing rate. However, then one cannot derive as the sum of two terms in (5), which is the main advantage that leads to our explicit analysis. Therefore, it seems that only stochastic simulation can be effective for non-additive settings. 1) Example 1: Consider a power generating system with years useful lifetime. We assume that the production level decreases linearly as the system degrades with time, and it is modelled by if if For convenience, we model the useful period as the one with positive output, when it is 0, the useful life is over. Therefore, the system should be replaced in case the optimal replacement was not performed before. On the contrary, in our example, the optimal replacement is performed much earlier than . Suppose that

FOR

TABLE II OPTIMAL TIME AND COST FOR DECREASING OUTPUT

THE

EXPONENTIALLY

This means that the corresponding lifetime follows the Weibull distribution, which is most often the case in practice for power generating systems. Let and 2.257 years (i.e., the mean-time to failure is 2 years). Let . The pictorial representation of the expected cost per unit of time for both the classical and proposed models as a function of the preventive maintenance interval 12 years is shown in Fig. 1. The corresponding values of interest can be found in Tables I and II. One can observe, as follows from the curves and the table, that the optimal point for the case of decreasing output is shifted to the left as compared with the optimal point for a constant or zero output. This perfectly corresponds to our reasoning above stating that . Note that when (exponential lifetime distribution), in the classical case, there is obviously no need for preventive maintenance, however in our case, the corresponding minimum can be still found due to degradation manifested by the decreasing .

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also does not change after this operation. Thus our repair is minimal both from the conventional stochastic point of view and from the proposed parametric one [2]. As the renewal cycle in this case is just , it is easy to derive the long run expected cost per unit of time for the described setting as [compare with (5)] (11) where is the mean number of minimal repairs in and, as previously, is the cost of replacement, whereas is now the cost of minimal repair . We are looking for the optimal that minimizes the cost function . It is clear that , whereas it is easy to show using the L’Hospital's rule that

Fig. 2. Cost functions for different values of parameter .

When is decreasing faster than linear, the effect of this degradation is “more prominent” which can be illustrated by comparing Fig. 1 with Fig. 2. The curves in Fig. 2 are obtained for the same parameters of Weibull distribution, whereas the output function has the following exponential form: . It can be also seen that, as expected, the effect of on the cost curve is increasing with the increase of parameter (in the sense that the minimums start to be more pronounced and shifted to the left). In practice, there can be other examples of deterioration in the output function that can be modelled by the considered in this paper approach. For instance, let a repairable system perform some tasks of a relatively short duration as compared with . Due to deterioration, the quality of performing these tasks is decreasing with time since the last repair (or replacement), which results in the increasing penalties. Denote the corresponding penalty function by (which can be considered as a negative output) that defines the penalty for failing the task at time with the maximal value . Denote the probability of performing the task at time since the last repair by , which is a decreasing function due to deterioration. Then the penalty function , can be defined as and our reasoning of the previous sections for obtaining the optimal can be applied. We are considering this setting for a specific case of the navigation-information system, where , is the function of the increasing navigation error . The results of this study will be published elsewhere. IV. PERIODIC REPLACEMENT POLICY Similar to the age replacement strategy, our -driven reasoning can be applied to the periodic replacement policies. Assume for simplicity and illustrative purposes that periodic replacements are performed at , whereas all failures that occur in between are instantaneously minimally repaired. By minimal repair we mean here that the function

Thus, depending on parameters (e.g., when ), the minimum can exist. Analyzing the derivative of it can be shown that, if the function

,

(12) obeys inequality , then the unique, optimal solution exists and is decreasing in and is increasing in . Obviously, when , the optimal solution is finite. Similar to (10), the optimal cost at is

The typical shape of the curves, e.g., for the Weibull lifetime distribution and is similar to that in Fig. 2. V. OPTIMAL REPLACEMENT WHEN THE OUTPUT STOCHASTIC PROCESS

IS A

We will now outline possible ways of generalization of the results of the previous section to the case when the output is a decreasing in the mean, positive stochastic process (13) is the increasing in the mean stochastic where process and . This generalization is rather straightforward as, formally, under some mild conditions, (4) holds and in (5) we just need to substitute the accumulated deterministic output by the expected accumulated output . Thus, in essence, we need only the mean of the process which is crucial for our approach and makes the corresponding modelling fairly simple. As is an increasing, degradation process, the assumption that is positive should be discussed. We are assuming it because we do not want to introduce a new failure mode and, therefore, to cross the zero level of the output. In most practical situations, distinct from the “core degradation processes,” e.g., as in the case of the growing crack in a material, the output usually does not


drop to some critical, inducing a failure, level. As was already mentioned, this is the difference of our approach with that employed while considering the condition-based maintenance. Remark 3: Note that Sections III and IV with deterministic output “do not need” the assumption of independence for the underlying processes of deterioration. However, for the stochastic case when 0 (or some other predetermined level of the output) is considered as the failure and the new failure mode arise, we need independence to proceed. Dependence will significantly complicate the derivations as, e.g., for defining the expected duration of the corresponding renewal cycle, the dependence structure between two modes should be given. However, if we avoid the new failure mode or consider situations when the negative output is relevant, then we also do not need independence. Positive can be modelled in different ways. First, the specific case of the corresponding stochastic process can be considered, namely, the path process of the form

is a function of a random parameter and . For instance, a good candidate for this path process would be , where is a non-negative (with support in random variable with the pdf . Then, obviously

where

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is the corresponding first passage time. Thus

(15) As the sample paths of the process are increasing, the integral in the numerator is also increasing in . The denominator is obviously decreasing and, therefore, is increasing and can be used for our purpose. The third option seems to be more reasonable from a practical point of view. Consider the initial stochastic process with parameters that guarantee sufficiently high probability that it will not reach during the lifetime described by the cdf with the mean . As an example, we can choose the stationary gamma process

where

is a decreasing positive function. Specifically, when is exponentially distributed with parameter which is a nice function for modelling deterioration in the output (see our example above). As the second option, we can use the corresponding conditional process on condition that the increasing process did not reach the level for each . Formally, let be a non-negative, monotonically increasing stochastic process with univariate pdf (for each ) and the corresponding cdf (for instance, the gamma process). As our goal is to obtain the expected values, only the defined univariate characteristics of are sufficient for this purpose. For excluding the possibility of values larger than , define the conditional univariate pdf [1]

(14) and the corresponding conditional univariate Cdf

Thus, the expected value of this process is linearly increasing. It follows from (13) that

and when , we arrive at the setting of Example 1. It is well known [11] that

where is the first passage time of the level , for the process . Therefore, our condition can be formulated as

This can be obviously achieved by the sufficiently small values of the “slope” . The last option obviously applies to deterministic output as well. In his case, the function should not necessarily be positive for all ; it should be positive in , where is sufficiently large as was the useful life in Example 1. Moreover, for the case of the gamma process, we have a linear decrease in as for the deterministic output in Example 1. VI. CONCLUSION

for the process the process

with values not exceeding . As is monotonically increasing, then

The title of our paper speaks for itself as we are considering the classical age replacement problem and generalizing it to the case when an operating system is described by the corresponding deterministic output function . Our output has a meaning of a reward in a small unit interval of time similar to the power in the power generating systems. The main assumption that describes degradation of our system (apart from the

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increasing failure rate) is that is a decreasing function. The accumulated reward in some interval of time, which of course is increasing, is the integral of in this interval of time. Another important assumption is that the process that results in the aging of a system's lifetime (e.g., manifested by increasing failure rate) and the process that leads to the decreasing are independent, which is reasonable at many instances in practice. For instance, the processes of misalignment of different parts of a system can result in decreasing output, whereas the failure occurs when one of the parts of the system experiences a hard failure. As expected, our results for age replacement model show that additional source of degradation decreases the optimal replacement time as compared with the case without output or with a constant in time output. Furthermore, the optimal replacement time can now exist and be finite when the failure time of our system is described by the exponential, non-aging distribution. Similar results hold for the periodic replacement briefly considered in the paper as well. As often for the ‘long run problems’, the case of a stochastic output is a rather straightforward generalization of the deterministic output case. This is due to the fact that only the expectation of stochastic output counts when obtaining stationary cost per unit of time. In our study, for simplicity and following practical considerations, we have assumed that the decreasing output does not constitute the corresponding failure mode. However, this case can also be considered as the optimal replacement strategy with competing risks. REFERENCES [1] M. Finkelstein and Z. Ludick, “On some steady-state characteristics of systems with gradual repair,” Rel. Eng. Syst. Safety, vol. 128, pp. 17–23, 2014. [2] M. Finkelstein, Failure Rate Modelling for Reliability and Risk. London, U.K.: Springer, 2008. [3] A. Lisniansky and G. Levitin, Multistate System Reliability. Theory With Applications. Singapore: World Scientific, 2003. [4] A. Lisnianski, I. Frenkel, and Y. Ding, Multistate System Reliability Analysis and Optimization for Engineers and Industrial Managers. London, U.K.: Springer, 2010. [5] L. Schu, L. Chen, J. Jin, J. Yu, F. Sun, and C. Wu, “Functional reliability simulation for a power-station's steam turbine,” Appl. Energy, vol. 80, pp. 61–66, 2005.

[6] R. E. Barlow and L. C. Hunter, “Optimum preventive maintenance policies,” Oper. Res, vol. 8, no. 1, pp. 90–100, 1960. [7] M. Finkelstein and J. H. Cha, Stochastic Modelling for Reliability. Shocks, Burn-in and Heterogeneous Populations. London, U.K.: Springer, 2013. [8] S. M. Ross, Stochastic Processes, 2nd ed. New York, NY, USA: Wiley, 1994. [9] T. Nakagawa, Advanced Reliability Models and Maintenance Policies. London, U.K.: Springer, 2008. [10] H. Z. Wang and H. Pham, Reliability and Optimal Maintenance. London, U.K.: Springer, 2006. [11] J. M. van Noortwijk, “A survey of the application of gamma processes in maintenance,” Rel. Eng. Syst. Safety, vol. 94, pp. 2–21, 2009. Maxim Finkelstein received the Ph.D. and Habilitation degrees from the Saint Petersburg Elektropribor Institute, Saint Petersburg, Russia, in 1979 and 1993, respectively. In 1998, he joined the University of the Free State, Bloemfontein, South Africa, as a Distinguished Professor. He is also a Visiting Professor with the Max Planck Institute for Demographic Research in Germany and with ITMO University, Saint Petersburg, Russia. He is a specialist in stochastic modeling, namely in mathematical theory of reliability. He is the author of more than 180 papers and five books on different aspects of reliability theory. His last book, coauthored with Prof. J. Hwan Cha, is titled Stochastic Modeling for Reliability: Shocks, Burn-in and Heterogeneous Populations (Springer, 2013). He is the editor of the South African Statistical Journal and an associate editor or a board member for a number of well-recognized international scientific journals in the field of stochastic modeling and reliability.

Mahmood Shafiee received the M.Sc. degree from Sharif University of Technology, Tehran, Iran, in 2006, and the Ph.D. degree from Iran University of Science and Technology, Tehran, in 2010. He is a Lecturer of engineering risk analysis with Cranfield University, Bedfordshire, U.K. His expertise is in the fields of reliability engineering, maintenance optimization, aging and degradation modeling, and probabilistic risk analysis. He has been awarded with several honors and educational scholarships at the national and international levels. He has also published more than 40 in top-tier journals, e.g., European Journal of Operational Research, Reliability Engineering and System Safety, Journal of Risk and Reliability, IIE Transactions, as well as many conference proceedings. He has been plenary speaker and program committee member of more than twenty international workshops and conferences.

Anselme N. Kotchap received the M.Sc. degree in mathematical statistics from the University of the Free State, Bloemfontein, South Africa, in 2011, where he is currently working toward the Ph.D. degree at the Department of Mathematical Statistics and Actuarial Science. His interests are in reliability modelling and, specifically, in optimal maintenance policies.